Given an integer n you need to calculate the number of distinct permutations {1, 2, 3, ..., n} such that the permutation represents a linear max heap.
In other words for each position from 1 to n: p[i] > p[2*i] and p[i] > p[2*i+1]
Example: Input: 4 Output: 3
I need help solving this problem, at least a hint please.
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Here
Is the runtime O(N) (assume we have preprocessed factorial and modular inverse of factorial) .
No, it's O(NlogN), because the modular inverse is calculated in O(log), atleast I don't know how to do it faster :)
If you know
, you can find out 
in O(1). 
can be computed in O(n).
Ohh yeah I just noticed it now
(n!) - 1 = (n + 1) * ((n + 1)!) - 1
Is this correct ?
Yes, that's the recurrence. Another way to do it, using one more array, is to precompute
for every i in 
, then 
. Here are some ideas to compute inverse efficiently.
I don't think it would be possible to precompute for all i because M can be very large (109 + 7)
In that problem M is 666013.
Yeah you are right . The preprocessing part takes O(NlogN) , and the part to calculate the number of heaps till N can be done in O(N) using the recurrence that you've posted . So overall it's O(NlogN) .
Thanks!